Fuzzy solution of the linear programming problem with interval coefficients in the constraints

A fuzzy concept of solving the linear programming problem with interval coefficients is proposed. For each optimism level of the decision maker (where the optimism concerns the certainty that no errors have been committed in the estimation of the interval coefficients and the belief that optimistic realisations of the interval coefficients will occur) another interval solution of the problem will be generated and the decision maker will be able to choose the final solution having a complete view of various possibilities.interval linear programming, fuzzy solution

Let's be Honest: An Optimal No-Regret Framework for Zero-Sum Games

We revisit the problem of solving two-player zero-sum games in the decentralized setting. We propose a simple algorithmic framework that simultaneously achieves the best rates for honest regret as well as adversarial regret, and in addition resolves the open problem of removing the logarithmic terms in convergence to the value of the game. We achieve this goal in three steps. First, we provide a novel analysis of the optimistic mirror descent (OMD), showing that it can be modified to guarantee fast convergence for both honest regret and value of the game, when the players are playing collaboratively. Second, we propose a new algorithm, dubbed as robust optimistic mirror descent (ROMD), which attains optimal adversarial regret without knowing the time horizon beforehand. Finally, we propose a simple signaling scheme, which enables us to bridge OMD and ROMD to achieve the best of both worlds. Numerical examples are presented to support our theoretical claims and show that our non-adaptive ROMD algorithm can be competitive to OMD with adaptive step-size selection.Comment: Proceedings of the 35th International Conference on Machine Learnin

Processor Allocation for Optimistic Parallelization of Irregular Programs

Optimistic parallelization is a promising approach for the parallelization of irregular algorithms: potentially interfering tasks are launched dynamically, and the runtime system detects conflicts between concurrent activities, aborting and rolling back conflicting tasks. However, parallelism in irregular algorithms is very complex. In a regular algorithm like dense matrix multiplication, the amount of parallelism can usually be expressed as a function of the problem size, so it is reasonably straightforward to determine how many processors should be allocated to execute a regular algorithm of a certain size (this is called the processor allocation problem). In contrast, parallelism in irregular algorithms can be a function of input parameters, and the amount of parallelism can vary dramatically during the execution of the irregular algorithm. Therefore, the processor allocation problem for irregular algorithms is very difficult. In this paper, we describe the first systematic strategy for addressing this problem. Our approach is based on a construct called the conflict graph, which (i) provides insight into the amount of parallelism that can be extracted from an irregular algorithm, and (ii) can be used to address the processor allocation problem for irregular algorithms. We show that this problem is related to a generalization of the unfriendly seating problem and, by extending Tur\'an's theorem, we obtain a worst-case class of problems for optimistic parallelization, which we use to derive a lower bound on the exploitable parallelism. Finally, using some theoretically derived properties and some experimental facts, we design a quick and stable control strategy for solving the processor allocation problem heuristically.Comment: 12 pages, 3 figures, extended version of SPAA 2011 brief announcemen

Predictive ability of problem-solving efficacy sources on mathematics achievement

This study examined the relationship between mathematics achievement and mathematics problem-solving efficacy sources. A cluster sample of 123 first year prospective teachers of a Philippine higher education institution responded to a 30-item problem-solving efficacy scales and took the teacher-made tests in Mathematics in the Modern World course; namely, Non-Routine Problem Solving and Natures and Numbers Pattern Tracing (NRPS-NNPT), Math Language and Symbols (MLS), and Data Management (DM). The research data was analyzed using Descriptive statistics, Pearson-r and Standard Multiple Regression. On the average, the respondents had satisfactory mathematics achievement. They reported a high level of social persuasion and somatic response and a low level of vicarious experience and mastery experience in mathematics problem-solving. Vicarious experience was directly associated with mastery experience while social persuasion and mastery experience were both inversely related to somatic responses. Among the four problem-solving efficacy sources, only social persuasion significantly predicted mathematics achievement specifically in the areas of NRPS-NNPT, MLS, and DM. Thus, becoming a trusted voice of encouragement and designing a persuasive and optimistic learning environment are highly recommended roles of schools to facilitate students’ mathematics achievement

Group composition : influences of optimism and lack thereof

Lesson video and video-stimulated post-lesson interviews were used to study the role of optimism in collaborative problem solving in a Grade 5/6 classroom for the purpose of informing group composition. This study focuses on the activity of two students who differed on the personal characteristic \u27optimistic orientation\u27. It examines how the presence or absence of an optimistic orientation to failures (Seligman, 1995) contributed to these students\u27 interactions with their groups and opportunities for collaborative creation of new knowledge. One group collaborated to develop mathematical knowledge that was new to each group member and the other group did not. These findings raise questions about how to group students who are not yet optimistic to enable collaborative activity, and how to build optimism.<br /

A Unified Distributed Method for Constrained Networked Optimization via Saddle-Point Dynamics

This paper develops a unified distributed method for solving two classes of constrained networked optimization problems, i.e., optimal consensus problem and resource allocation problem with non-identical set constraints. We first transform these two constrained networked optimization problems into a unified saddle-point problem framework with set constraints. Subsequently, two projection-based primal-dual algorithms via Optimistic Gradient Descent Ascent (OGDA) method and Extra-gradient (EG) method are developed for solving constrained saddle-point problems. It is shown that the developed algorithms achieve exact convergence to a saddle point with an ergodic convergence rate $O(1/k)$ for general convex-concave functions. Based on the proposed primal-dual algorithms via saddle-point dynamics, we develop unified distributed algorithm design and convergence analysis for these two networked optimization problems. Finally, two numerical examples are presented to demonstrate the theoretical results